Two conductors have the same resistance at 0^ C but their...

Let $R_0$ be the initial resistance of both conductors $\therefore \quad$ At temperature $\theta$ their resistance will be, $R_1=R_0\left(1+\alpha_1 \theta\right)$ and $R_2=R_0\left(1+\alpha_2 \theta\right)$ for, series combination, $\mathrm{R}_{\mathrm{s}}=\mathrm{R}_1+\mathrm{R}_2 R_{s 0}\left(1+\alpha_s \theta\right)$=R_0\left(1+\alpha_1 \theta\right)$+R_0\left(1+\alpha_2 \theta\right)$ where $\mathrm{R}_{\mathrm{s} 0}=\mathrm{R}_0+\mathrm{R}_0=2 \mathrm{R}_0$ \therefore \quad 2 $\mathrm{R}_0\left(1+\alpha_s \theta\right)=2 \mathrm{R}_0+\mathrm{R}_0$ \theta\left(\alpha_1+\alpha_2\right)$or$\quad \alpha_s=\frac{\alpha_1+\alpha_2}{2}$for parallel combination, R_p=$\frac{R_1 R_2}{R_1+R_2}$where,$R_{p 0}=\frac{R_0 R_0}{R_0+R_0}=\frac{R_0}{2} \therefore \quad \frac{\mathrm{R}_0}{2}$\left(1+\alpha_{\mathrm{p}} \theta\right)=\frac{\mathrm{R}_0^2\left(1+\alpha_1 \theta+\alpha_2 \theta+\alpha_1 \alpha_2 \theta\right)}{\mathrm{R}_0\left(2+\alpha_1 \theta+\alpha_2 \theta\right)}$ as $\alpha_1$ and $\alpha_2$ are small quantities $\therefore \quad \alpha_1 \alpha_2$ is negligible or $\quad \alpha_{\mathrm{p}}=\frac{\alpha_1+\alpha_2}{2+\left(\alpha_1+\alpha_2\right) \theta}=\frac{\alpha_1+\alpha_2}{2}\left[1-\left(\alpha_1+\alpha_2\right) \theta\right]$ as $\left(\alpha_1+\alpha_2\right)^2$ is negligible $\therefore \quad \alpha_p=\frac{\alpha_1+\alpha_2}{2}$